5,552 research outputs found
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
We present a deterministic way of assigning small (log bit) weights to the
edges of a bipartite planar graph so that the minimum weight perfect matching
becomes unique. The isolation lemma as described in (Mulmuley et al. 1987)
achieves the same for general graphs using a randomized weighting scheme,
whereas we can do it deterministically when restricted to bipartite planar
graphs. As a consequence, we reduce both decision and construction versions of
the matching problem to testing whether a matrix is singular, under the promise
that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm
for bipartite planar graphs. This improves the earlier known bounds of
non-uniform SPL by (Allender et al. 1999) and by (Miller and Naor 1995,
Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a
deterministic parallel algorithm for constructing a perfect matching in
non-bipartite planar graphs, which has been open for a long time. Our
techniques are elementary and simple
Embedding into bipartite graphs
The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher,
Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any
, every balanced bipartite graph on vertices with bounded degree
and sublinear bandwidth appears as a subgraph of any -vertex graph with
minimum degree , provided that is sufficiently large. We show
that this threshold can be cut in half to an essentially best-possible minimum
degree of when we have the additional structural
information of the host graph being balanced bipartite. This complements
results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and
Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding
minimum degree threshold for -factors, with and fixed.
Moreover, it implies that the set of Hamilton cycles of is a generating
system for its cycle space.Comment: 16 pages, 2 figure
Streaming Verification of Graph Properties
Streaming interactive proofs (SIPs) are a framework for outsourced
computation. A computationally limited streaming client (the verifier) hands
over a large data set to an untrusted server (the prover) in the cloud and the
two parties run a protocol to confirm the correctness of result with high
probability. SIPs are particularly interesting for problems that are hard to
solve (or even approximate) well in a streaming setting. The most notable of
these problems is finding maximum matchings, which has received intense
interest in recent years but has strong lower bounds even for constant factor
approximations.
In this paper, we present efficient streaming interactive proofs that can
verify maximum matchings exactly. Our results cover all flavors of matchings
(bipartite/non-bipartite and weighted). In addition, we also present streaming
verifiers for approximate metric TSP. In particular, these are the first
efficient results for weighted matchings and for metric TSP in any streaming
verification model.Comment: 26 pages, 2 figure, 1 tabl
The critical Z-invariant Ising model via dimers: the periodic case
We study a large class of critical two-dimensional Ising models namely
critical Z-invariant Ising models on periodic graphs, example of which are the
classical square, triangular and honeycomb lattice at the critical temperature.
Fisher introduced a correspondence between the Ising model and the dimer model
on a decorated graph, thus setting dimer techniques as a powerful tool for
understanding the Ising model. In this paper, we give a full description of the
dimer model corresponding to the critical Z-invariant Ising model. We prove
that the dimer characteristic polynomial is equal (up to a constant) to the
critical Laplacian characteristic polynomial, and defines a Harnack curve of
genus 0. We prove an explicit expression for the free energy, and for the Gibbs
measure obtained as weak limit of Boltzmann measures.Comment: 35 pages, 8 figure
Grafalgo - A Library of Graph Algorithms and Supporting Data Structures (revised)
This report provides an (updated) overview of {\sl Grafalgo}, an open-source
library of graph algorithms and the data structures used to implement them. The
programs in this library were originally written to support a graduate class in
advanced data structures and algorithms at Washington University. Because the
code's primary purpose was pedagogical, it was written to be as straightforward
as possible, while still being highly efficient. Grafalgo is implemented in C++
and incorporates some features of C++11.
The library is available on an open-source basis and may be downloaded from
https://code.google.com/p/grafalgo/. Source code documentation is at
www.arl.wustl.edu/\textasciitilde jst/doc/grafalgo. While not designed as
production code, the library is suitable for use in larger systems, so long as
its limitations are understood. The readability of the code also makes it
relatively straightforward to extend it for other purposes
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